From Trees to Loops and Back

TL;DR

This paper demonstrates that one-loop scattering amplitudes in supersymmetric Yang-Mills theories can be equivalently computed using MHV diagrams or Feynman diagrams, establishing covariance and matching discontinuities.

Contribution

It provides a general proof of covariance for one-loop non-MHV amplitudes from MHV diagrams and shows their equivalence to standard Feynman diagram calculations.

Findings

Discontinuities of amplitudes match between MHV and Feynman methods.

Universal one-loop splitting functions derived for supersymmetric Yang-Mills.

Application of Feynman's Tree Theorem to one-loop MHV and Feynman diagrams.

Abstract

We argue that generic one-loop scattering amplitudes in supersymmetric Yang-Mills theories can be computed equivalently with MHV diagrams or with Feynman diagrams. We first present a general proof of the covariance of one-loop non-MHV amplitudes obtained from MHV diagrams. This proof relies only on the local character in Minkowski space of MHV vertices and on an application of the Feynman Tree Theorem. We then show that the discontinuities of one-loop scattering amplitudes computed with MHV diagrams are precisely the same as those computed with standard methods. Furthermore, we analyse collinear limits and soft limits of generic non-MHV amplitudes in supersymmetric Yang-Mills theories with one-loop MHV diagrams. In particular, we find a simple explicit derivation of the universal one-loop splitting functions in supersymmetric Yang-Mills theories to all orders in the dimensional regularisation parameter, which is in complete agreement with known results. Finally, we present concrete and illustrative applications of Feynman's Tree Theorem to one-loop MHV diagrams as well as to one-loop Feynman diagrams.